Some complexity bounds for dynamic logics

The class of so called Adian´s structures with pairwise different exponents is considered. It is known that both deterministic dynamic logic (DDL) and context-free DDL (CF-DDL) have an unwind property in every structure Γ in the class; thus they are equivalent in Γ to first-order logic. None the less, it turns out that these three logics have different complexity bounds in the class. The main result is to prove polynomial upper bounds for DDL formulas. As a corollary, the authors find the DDL and CF-DDL are unequivalent to one another in the class. The proof remains valid even in the presence of elementary tests and rich tests